What does the term "Speed" mean in a speaker?

Most speakers are not time cohesive and nobody seems to care or notice.

Just because nobody seems to care does not mean it does not matter.  Just because if something has not been proven, it does not mean that it's not true.  

I agree with most of the theories above for sure.  I also agree that the room acoustics have a lot to do with frequency dispersion and speed. To much absorption and or diffusion will have a profound effect on the fastest speaker production.
It is unfortunate fact for all of this as it really difficult to find the perfect combination without serious experimentation and or expense. I am fortunate to own full ribbon driver speakers and have had all types before them......they are easily the fastest I have experienced because of driver mass and height of the panel allowing driver radiation accuracy. In addition, I have experimented with acoustic treatments extensively (both absorption and diffusion) and feel they do pretty well with minimal treatment accept for my room's obvious modes. Nothing will be perfect so you have to choose the combination that pleases you most in the overall selection of components, speakers, and room acoustics. Everyone has different thoughts in perceived musical reproduction.

Best of luck,
gwalt

Speed = Rise and Decay

I enjoy a fast midrange/woofer/subwoofer and cabinet design, as it sounds crisp and accurate. 

Tweeters are generally fast. I enjoyed the Focal Tc120/90 series tdxt with the vented pole piece, as you could build a rear enclosure and affect the sound/speed of the tweeter like other drivers in the system. 

Listen to the Bass Line in Glad Tidings by Van Morrison.....

Does the whole song sound in sync or can you sense some lag ...

Speed as a function...

Small floorstander, Castle Conway III, intel Nuc, Benchmark Dac pre, Mystere PA, you want ‘speed’, transient attack, dynamics with decay, openness?  Try Captains Audio Envy cables.  I’ve a pr. of the Studio Prestige rca, a usb, & a balanced cable.  Remarkable spacious detailed resolution, the fastest cable I’ve heard.  Lightweight, easy to handle, affordable, speed.   

Example... the Castles r bottom ported adjustable thru spacers to various room sizes.  With AE interconnects, ports wide open, they sound like sealed boxes, absolutely no overhang, spacious, neutral, dynamic, textured speed unleashed thru the speakers carbon fiber drivers.

Addictive.

Same thru a pr. of 1SC’s.  AE interconnects r FAST, and one can easily appreciate such thru the additional frequency range moving beyond the monitors.  Fun too.  Refined, cohesive, spacious, they convey a distinct ‘house sound’ that reminds of a good modern tube amp, or a great set of headphones, the small Castles sound like lovely dark electrostatics with better dynamics.  
Speed indeed...